Question

It is well-known that $BU^+$ is homotopy equivalent to an infinite loop space where $U$ is the limit of the unitary groups $U(n)$ for $n \rightarrow \infty$ and $+$ denotes Quillen's Plus construction.

On the other hand there is a tool to detect infinite loop spaces by the action from $E\Sigma_p$ ($\Sigma_p$ is the symmetric group).

My question is: how does the map $E\Sigma_p \times_{\Sigma_p} BU^p \rightarrow BU$ looks like. Is it possible to write it down.

Answer 1

There are a number of equivalent ways of seeing $BU\times Z$ or $BU$ as an infinite loop space. The description that Neil gives is the action of the linear isometries operad $\mathcal{L}$ (complex version) on $BU$. A large number of related spaces have such a structure, as explained in the first section of the first chapter of "$E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra", SLN 577 (1977). Morever, the Bott maps are maps of $\mathcal{L}$-spaces, which ties in Bott's original proof that $BU$ is an infinite loop space. The fact that $\coprod_n U(n)$ is a permutative category gives a quite different operad action on $\coprod_n BU(n)$, whose associated infinite loop space is $BU\times \mathbf{Z}$. It is not obvious a priori how to compare the infinite loop structures on these two models. The question is resolved in this and related examples (e.g. $BTop$) in my paper "The spectra associated to $\mathcal{I}$-monoids". Math. Proc. Camb. Phil. Soc. 84(1978), 313--322.

Answer 2

Firstly, no plus construction is needed here. The plus construction is used to kill a perfect subgroup of the fundamental group, but $\pi_1(BU)$ is already trivial.

Next, put $\mathcal{V}=\bigoplus_{n=0}^\infty\mathbb{C}$, and equip this with the standard Hermitian inner product. Let $\mathcal{V}_n$ be the obvious copy of $\mathcal{C}^n$ in $\mathcal{V}$. Let $B$ denote the space of subspaces $\mathcal{W}\leq\mathcal{V}\oplus\mathcal{V}$ such that $\mathcal{W}\cap(\mathcal{V}\oplus 0)$ has finite codimension in $\mathcal{W}$, and also the same finite codimension in $\mathcal{V}\oplus 0$. This is the union of subspaces $$ B(n) = \{\mathcal{W}\in B : \mathcal{V}_n^\perp\oplus 0 \leq\mathcal{W}\leq\mathcal{V}\oplus\mathcal{V}_n, \dim(\mathcal{W}/(\mathcal{V}_n^\perp\oplus 0))=n\}$$ Now $B(n)$ is homeomorphic to the Grassmannian $G_n(\mathbb{C}^{2n})$, and using this we see that $B$ is a model for $BU$.

Now let $E$ denote the space of inner-product preserving linear maps from $\mathcal{V}^k$ to $\mathcal{V}$. This has an evident action of $\Sigma_k$, which is free because inner-product preserving maps are always injective. It is a standard fact that $E$ is also contractible, so it is a model for $E\Sigma_k$.

Now suppose we have elements $\mathcal{W}_1,\dotsc,\mathcal{W}_k\in B$ and a map $f\in E$. We then have a subspace $$\bigoplus_i\mathcal{W}_i\leq\bigoplus_{i=1}^k (\mathcal{V}\oplus\mathcal{V}) \simeq \left(\bigoplus_{i=1}^k\mathcal{V}\right)\oplus \left(\bigoplus_{i=1}^k\mathcal{V}\right). $$ We can apply $f\oplus f$ to this to get a point $\gamma(f;\mathcal{W}_1,\dotsc,\mathcal{W}_k)\in B$. This construction gives the map $\gamma:E\times_{\Sigma_k}B^k\to B$ that you need.